Pandigital and penholodigital prime digit-sum conjecture

For a base-$b$ integer $n$, let $s_b(n)$ denote the sum of its base-$b$ digits. A pandigital prime (respectively, penholodigital prime) is a prime whose base-$b$ representation contains every digit $0,1,\ldots,b-1$ (respectively, every nonzero digit $1,\ldots,b-1$) at least once.

Pandigital and penholodigital prime digit-sum conjecture. For $b>3$, the smallest pandigital prime or penholodigital prime $n$ satisfies $s_b(n)=b(b-1)/2+2$ if $b$ is of the form $4k+3$, and satisfies $s_b(n)=b(b-1)/2+1$ otherwise.

The preceding theorems establish corresponding lower bounds, with the stronger bound in bases of the form $4k+3$. Numerical experiments are cited as evidence, but the conjecture is not resolved in the supplied text.